Knots (with METAFONT)

The knot drawings were first shown on the page Knoten - Seminar in 2021/2022.

Another display of essentially the same drawings is on the page Knots (with METAFONT) - Gallery.

Here is a particular example:

knot 5₂

Click on the images for high resolution versions.

The pictures were created 1995 using METAFONT, see allknots.pdf and Source files (1995).

These were experiments inspired by Exercise 13.8 in The METAFONTbook Zbl 0609.68003. I stopped after saving allknots.pdf (actually allknots.dvi) on July 6, 1995 at 4:11 AM.

There are 4 knot diagrams with 9 crossings in allknots.pdf. One is a non-standard drawing of knot 8₁₈ (Torusknoten (4,3)), the others are

knot 9₂₂knot 9₂₉knot 9₄₀

Cf. Appendix D in Burde, Zieschang. Knots. 1985 MR 808776, Zbl 0568.57001:

Screenshots from Burde-Zieschang

Originally all over/under-crossings were drawn alternating. This is incorrect for the torus knots (3,4), (4,3), (6,11), albeit one may prefer the alternating version in a non-mathematical context. The corresponding images have now (March 2026) correct crossings (unless marked with *).

On the page Knots (with METAFONT) - Gallery most torus knots are rectified to have a vertical axis of symmetry, cf. Drawing torus knots on that page.

Quite a few torus knots are depicted here: Category:Torus knots and links (Wikimedia Commons)

The images linked in the following list have high resolutions. The numbers correspond to page numbers in allknots.pdf.

The names "knot C_k", "Knoten C/k" refer to the Alexander-Briggs notation, see Appendix C in Rolfsen, Knots and Links. 1976 MR 515288, Zbl 0339.55004.

torus knot (2,3) - knot 3₁
- Torusknoten (2,3) [thin version]
- Common names: Kleeblattschlinge, trefoil knot (Wikipedia)
- (2,3)_torus_knot.svg (Wikimedia)
knot 4₁
- Viererknoten (four-knot) [thin version]
- By some called Achterknoten, figure-eight knot (Wikipedia).
- The names Viererknoten, four-knot appear in
  - Burde, Zieschang. Neuwirthsche Knoten und Flächenabbildungen. 1967 MR 229229, Zbl 0171.22402.
  - Burde, Zieschang. Knots. 1985 MR 808776, Zbl 0568.57001.
  - Burde, Zieschang. Knots (second edition). 2003 MR 1959408, Zbl 1009.57003.
  - Burde, Zieschang, Heusener. Knots (third, fully revised and extended edition). 2014 MR 3156509, Zbl 1283.57002.
torus knot (2,5) - knot 5₁
- Torusknoten (2,5) [thin version]
- Other name: cinquefoil knot (Wikipedia)
- (2,5)_torus_knot.svg (Wikimedia)
knot 5₂
- "Knoten 5/2 (Versuch zu 9/29)" [thin version]
- "Knoten 5/2 (Versuch zu 9/29)" (gespiegelt) [thin version] (mirrored)
- - well, ok for a first try (as for the drawing)
- This is the twist knot with three half twists, see three-twist knot (Wikipedia).
- Apparently this is an unusual way of drawing knot 5₂ (I couldn't find a similar image elsewhere).
torus knot (2,7) - knot 7₁
- Torusknoten (2,7) [thin version]
- (2,7)_torus_knot.svg (Wikimedia)
torus knot (3,4) - knot 8₁₈
- Torusknoten (3,4) [thin version]
- Torusknoten (3,4) [thin version] (mirrored)
- *Torusknoten (3,4) [thin version] (with alternating crossings)
- (3,4)_torus_knot.svg (Wikimedia)
torus knot (4,3) - knot 8₁₈ by a diagram with 9 crossings
- Torusknoten (4,3) [thin version]
- Torusknoten (4,3) [thin version] (mirrored)
- *Torusknoten (4,3) [thin version] (with alternating crossings)
- I couldn't find the drawing elsewhere.
"Knoten 9/22" [thin version] - a strange knot (I wasn't satisfied with this one)
"Knoten 9/29" [thin version] - looks better (my favorite)
"Knoten 9/40" [thin version] - not bad
torus knot (6,11) - 55 crossings
- Torusknoten (6,11) [thin version]
- Torusknoten (6,11) [thin version] (mirrored)
- *Torusknoten (6,11) [thin version] (with alternating crossings) - another favorite
- I couldn't find the drawing elsewhere.
torus knot (6,11) with numbered crossings is a scan from page 12 of allknots.pdf.

Source files (1995)

The file allknots.txz (xz compressed tar archive) contains the 1995 MF files and a script to build allknots.pdf.

In 2009 I created png images with ImageMagick taking data from MF log files. The cutting at crossings was done with ImageMagick. See directory 2009 in allknots.txz.

In 2026 I added the possibility to draw torus knots with correct over/under-crossings (fortunately there was not much to do).